Relativistic magnetohydrodynamics in one dimension
We derive a number of solution for one-dimensional dynamics of relativistic magnetized plasma that can be used as benchmark estimates in relativistic hydrodynamic and magnetohydrodynamic numerical codes.
First, we analyze the properties of simple waves of fast modes propagating orthogonally to the magnetic field in relativistically hot plasma. The magnetic and kinetic pressures obey different equations of state, so that the system behaves as a mixture of gases with different polytropic indices. We find the self-similar solutions for the expansion of hot strongly magnetized plasma into vacuum.
Second, we derive linear hodograph and Darboux equations for the relativistic Khalatnikov potential, which describe arbitrary one-dimensional isentropic relativistic motion of cold magnetized plasma and find their general and particular solutions. The obtained hodograph and Darboux equations are very powerful: system of highly non-linear, relativistic, time dependent equations describing arbitrary (not necessarily self-similar) dynamics of highly magnetized plasma reduces to a single linear differential equation.
Expansion of plasma into vacuum is a basic problem in fluid mechanics that has a wide range of applications from heavy nuclei collisions to astrophysics. In nuclear physics, Belenkij and Landau Belenkij and Landau (1956) used the hydrodynamical approach to study multiÐparticle production in heavy ion collisions. A head-on collision of two highly relativistic nuclei creates a relativistically-compressed hot layer of quarkÐgluon plasma that expands quasi-one-dimensionally Rischke et al. (1995). On a very different scale, a wide variety of astrophysical objects like jets from Active Galactic Nuclei (AGN) Blandford (2002), Gamma Ray Burst (GRB) Lyutikov (2006a) and pulsar winds and magnetospheres of a special type of neutron stars - magnetars Thompson and Duncan (1993)- may contain relativistic strongly magnetized plasma, in which the energy density of the magnetic field dominates over the matter energy density (including kinetic and thermal energies), . During explosions the strongly magnetized plasma created by the central source suddenly expands into a surrounding low density medium. In case of magnetar flares, the initial dissipation event creates relativistically hot, strongly magnetized fireball, somewhat analogous to Solar coronal mass ejections Lyutikov (2006b); the fireball accelerates to relativistic velocities Palmer et al. (2005). In long GRBs, when a hot magnetically dominated jet reaches the surface of the star it breaks into low density medium Lyutikov (2006a); Tchekhovskoy et al. (2008). Similar dynamics may occur in non-stationary outflows in AGNs as well Lyutikov and Lister (2010).
In all the above mentioned cases, it is expected that at some distances from the source, the magnetic field is dominated by the toroidal component, while motion is preferentially radial. Thus, velocity is nearly perpendicular to magnetic field - this type of motion is sometimes called transverse magnetohydrodynamics. Equations of transverse MHD reduce to fluid equations, but with a complicated equation of state Landau and Lifshitz (1960). Qualitatively, such plasma behaves like a mixture of fluids with different adiabatic indices, for the magnetic field, and some for the kinetic pressure, usually taken to be . Thus, many results of fluid dynamics, which assume a single adiabatic index, become invalid.
Numerical investigation of these phenomena requires the use of relativistic MHD codes that can handle high magnetization and high kinetic pressures exceeding the rest mass density. Exact, explicit non-linear solutions of fluid equations, and especially relativistic MHD equations, are rare. Yet they are important for benchmark estimates of the overall dynamical behavior in numerical simulations of relativistic flows and strongly magnetized outflows in particular. In §II we find analytical expressions for self-similar expansion into vacuum of a hot magnetized plasma, considering both the Newtonian and relativistic cases with arbitrary ratios of kinetic and magnetic pressures to rest mass density.
In addition, at later times, when the whole initial state of plasma is affected by the expansion, the expansion dynamics becomes non-self-similar. A classical related problem is then the expansion of a slab of finite length.
Arbitrary isentropic one-dimensional motion of a fluid is fully integrable. Mathematically this is achieved by exchanging the independent variables and dependent variables . As a result a system of two non-linear equations (of mass conservation and Euler’s law plus assumed isentropic equation of state) is reduced to a single linear equation for the (Legendre-transformed) Bernoulli (Khalatnikov) potential (e.g. Landau and Lifshitz, 1959). This is always possible if the coefficients in these equations do not depend explicitly on time and coordinate. In the case of isentropic fluid, the equation of state can be correspondingly inverted. This transformation is called the hodograph transformation Landau and Lifshitz (1959). In §IV we perform a hodograph transformation for relativistic cold magnetized plasma and derive the corresponding hodograph and Darboux equations.
Ii Relativistic expansion of hot magnetized plasma into vacuum
Consider a relativistic one-dimensional flow of hot magnetized fluid along direction, carrying magnetic field in perpendicular direction (so-called transverse MHD) The governing of the equations are (Landau and Lifshitz, 1959)
where is the proper (plasma frame) magnetic field divided by , and are fluid’s velocities in terms of the speed of light and Lorentz factors, is the proper energy density, is proper mass density and is proper enthalpy. The adiabatic index for the kinetic pressure is assumed to be .
Let us consider a semi-space occupied by homogeneous hot plasma with density , magnetic field and pressure . The initial state is assumed to be homogeneous. (At time a barrier located at is removed; plasma starts expanding in the positive direction, while a rarefaction wave is launched in the negative direction. We introduce two dimensionless parameters describing the initial plasma magnetization and the ratio of kinetic to magnetic pressures:
Parameter Kennel and Coroniti (1984) measures the importance of relativistic effects of magnetic fields: for the Alfvén velocity approaches the speed of light. Parameter is the conventional plasma beta parameter: the ratio of kinetic and magnetic pressures.
The relevant speed of propagation of disturbances is the fast magnetosonic speed (Harris, 1957)
where is Alfvén velocity and is sound speed. Note that for , the fast speed is independent of and equals . In this case the contribution of kinetic pressure to total pressure is compensated by its contribution to effective mass density.
For transverse MHD (when the motion is perpendicular to the magnetic field), the induction equation and matter conservation require . Assuming that all the quantities depend on the self-similar variable , Eqns. (1) give
where and primes denote derivative with respect to (note, that if the initial state is not homogeneous, the self-similar solutions may depend on combination ).
Eliminating pressure and magnetic field in the initial state in favor of and , , , , we find
Changing to Doppler factors
It is more convenient to change to a new variable and redefine parameter (in this case the resulting equations (10-11) depend only on one parameter ; also, in the cold limit becomes a four-velocity of Alfvén waves Lyutikov (2010)). In terms of variables , Eqns (8-9) become
Previously, Lyutikov Lyutikov (2010) found simple analytical solution of Eqns. (10-11) for simple waves in cold magnetized plasma expanding into vacuum. If initially the plasma is at rest, and the Alfvén velocity in the unperturbed medium is given by corresponding Doppler factor , in the limit , Eqns. (10-12) have solutions
In case of hot plasma, , Eq. (10) can be resolved for
This gives a general solution for the Doppler factor in terms of initial parameters and local density .
Eq. (11) then becomes
or, using the explicit form of , Eq. (13),
Changing independent variables , we find:
Eq. (16) gives a general solution for simple fast waves in relativistic magnetized fluid with arbitrary ratios of magnetic and kinetic pressure to rest mass, with kinetic part of the pressure obeying adiabatic law with . Eq. (16) expresses implicitly the density in terms of the self-similar variable .
The lower limit of integration in Eq. (16) can be found from the conditions on the front characteristics propagating into undisturbed plasma. The rarefaction wave propagates into the undisturbed medium with the local fast velocity (see Eq. (3)):
This corresponds to
(note the signs of velocities in the definition of : the rarefaction wave propagates in the direction opposite to the flow). At the front of the rarefaction wave . Thus, a particular solution corresponding to expansion into vacuum of a medium initially at rest is
in agreement with Lyutikov (2010).
For zero kinetic pressure, , relations (19-20) imply that the front of the rarefaction wave is located at , while for large kinetic pressure, , the front of the rarefaction wave is located at . The vacuum interface in the limit approaches the speed of light.
For a fixed values of , the increase of kinetic pressure leads to larger values of negative (smaller absolute values of ) due to effective increase in plasma inertia. For , the front of the rarefaction wave is located at fixed for any . Finally, we note that the Riemann invariants in a transverse relativistic MHD, does not have a representation in simple functions.
Iii Self-similar expansion of unmagnetized relativistic fluid
In this section we simplify the above relations for the case of simple waves in relativistic unmagnetized fluid. Adopting a polytropic EoS, we will calculate simple waves in plasma with adiabatic index of , yet for a finite ratio , without assuming that the speed of sound equals . This is an important step since most codes use a single value of adiabatic index, but for finite ratios .
For a finite ratio of , the speed of sound equals
while Eqns. (6) become
In case of negligible density , this immediately gives the ultra-relativistic limit ,
The front of the rarefaction wave is located at , while on the vacuum side expansion proceeds with the speed of light.
In the ultra-relativistic limit the flow lines are given by
The characteristics satisfy , which gives
(the other characteristics are straight lines).
Which can be integrated
The constant of integration in Eq. (29) can be found from the condition that at the front of the rarefaction wave, propagating with velocity and located at the fluid is at rest, . This gives
Iv Non-similar expansion of cold magnetized plasma: relativistic hodograph and Darboux equations
Relativistic hodograph transformation was derived by Belenkij & Landau Belenkij and Landau (1956) for polytropic fluid. Here we first re-derive the corresponding equation for cold magnetized plasma and then transform it to a normal form, where Riemann invariants are taken as independent variables. In case of regular fluids the internal energy and the corresponding hodograph equations are defined in term of temperature, which is zero in the case of cold magnetized fluid. As we will see below, the role of temperature is taken by the proper enthalpy.
The relativistic hodograph transformation is achieved by introducing the Khalatnikov potential (Landau and Lifshitz, 1959)
In the non-relativistic limit the corresponding equations are the condition on potential flow and the Bernoulli equation. Below we drop the tilde sign over the enthalpy: is then the enthalpy per unit mass:
The differential of the Khalatnikov potential is
Next we employ Legendre transform of with respect to variables . The transformed potential becomes
Variables and are then expressed from the potential as
These equations can be modified if we introduce rapidity :
Comparing with Ref. Belenkij and Landau (1956) Eq. (4.12), we see that in cold magnetized plasma proper enthalpy plays a role of a temperature, while the speed of sound – Alfvén velocity - is given by
The relativistic hodograph equation is then obtained from the continuity equation by transforming to independent variables and .
This is relativistic hodgoraph equation for one-dimensional motion of cold magnetized plasma. Eq. (38) reduces to the one obtained by Belenkij and Landau for hot fluid with a substitution . A general separable solution of Eq. (38) is
where, we remind, is the rapidity, , and is the hypergeometric function.
are corresponding Doppler factors.
Using Eqns (41), the Riemann invariants become
Thus, the rapidity and the proper enthalpy are
Changing the independent variables in the hodograph equation (38), we derive the relativistic Darboux equation for the Khalatnikov potential as a function of the Riemann invariants:
Using constancy of the Riemann invariants on the characteristics, one finds (cf., Eq, (45))
System (45) can be written as a single equation for time
This is the relativistic analogue of the Darboux equation for the time variable. Solutions of Eqs. (46-45) give time and spacial coordinate as functions of two Riemann invariants. These solutions can then be inverted for .
V 1D expansion of a slab of gas: solutions of the hodograph equation
Consider a slab of gas initially at rest, occupying region and expanding into vacuum . At there is an impenetrable wall, Fig. 4. In the initial state the Riemann invariants are where is the sound (Alfvén ) velocity in the initial state. The boundary conditions for this problem are (Landau and Lifshitz, 1959, problem after parag. 105): zero velocity at the wall and constancy of the first Riemann invariant on the characteristics that leaves the wall at the moment of reflection:
Numerical solutions for Alfvén and sound speeds are found and plotted in both the non-relativistic (as discussed in Appendix B) and relativistic case. In both cases finding a numerical solution involves inverting a pair of functions of the Riemann invariants, , to obtain functions of position and time giving the Riemann invariants, . In the non-relativistic case the initial functions are functions represented by Eq.’s (74) and (75) whereas in the relativistic case these functions are determined using numerical methods. The relevant quantities, Alfvén and sound speed, can then be expressed as functions of position and time. The numerical solutions in both relativistic and non-relativistic cases involve separate calculations for the regions of expansion affected by the reflection of the initial rarefaction wave at the position of the wall, , and the regions unaffected by this reflection (as well as the boundary between these regions). The former region will be referred to as the “complex” region and latter as the “simple” region.
B. Numerical Solutions of Relativistic Expansion
Initial conditions in the relativistic case are given, by initial Alfvén speed initial velocity , the plasma occupies the region with an impenetrable wall at and the plasma expands into vacuum at . The initial conditions are such that reflection of the initial rarefaction wave occurs at . The Initial Alfvén speed, , was selected as a speed somewhat near the speed of light that gives convenient initial values for the Riemann invariants: .
In the case of relativistic expansion, no simple analogue to Eq. (74) that satisfies Eq. (46) and gives time as a function of the Riemann invariants in the complex wave region is readily available. Therefore, in the complex region, time must be numerically calculated as a function of the two Riemann invariants, . The first step in this calculation is accomplished by numerically solving for a function satisfying the adjoint differential operator corresponding to Eq. (73) subject to boundary conditions as outlined in (Courant and Hilbert, 1953, Ch. 5, Sec. 2, (Eq.’s 3 and 3’)), obtaining a solution . In the present calculations this function is solved numerically after specifying boundary values based on and . Exploiting the symmetry of the Riemann’s function, (Courant and Hilbert, 1953, Ch. 5, Sec. 2, (Eq.’s 3 and 3’)),, one obtains an expression for time given as a function of satisfying (46) :
The initial value of both Riemann invariants, , corresponds to the initial conditions of and . Position can then be calculated as a function of the Riemann invariants by integrating by parts Eq. (45).
As a next step towards a numerical solution, is calculated as a function of and time by identifying the values of corresponding to a specified time through an interpolation method similar to the one used in the non-relativistic case, resulting in the calculation of a function . In order to determine the numerical solutions in the complex region for a given time, a collection of points of the form , where we have emphasized only a parametric dependence on time, are calculated based on a number of values in the range . This range represents the values in the complex region and represents the minimum value encountered at the point subject to the condition . This condition is based on the requirement that and hence . Having obtained a sufficient collection of points, the solutions
can be numerically calculated by interpolation. The functions and in the simple region of the expansion are known analytical functions of and , more specifically of a single variable , given in Ref. Lyutikov (2010) as
The boundary, , between the simple and complex regions of expansion is determined as a function of time in terms of by the equation Lyutikov (2010)
Previously, the problem of non-self-similar expansion of magnetized plasma was considered numerically in Ref. Granot et al. (2011). Contrary to the initial claim in Ref. Granot et al. (2011), the presence of the wall is detrimental to acceleration, as can be seen from Fig. 4. Magnetic pressure-driven acceleration proceeds most efficiently while a given fluid element is causally disconnected from the wall, during the self-similar stage discussed in Ref. Lyutikov (2010).
In this paper we derived a number of analytical results for one-dimensional expansion of magnetized gas into plasma. First, we found the self-similar expansion into vacuum of a hot magnetized plasma. In this case the total pressure has contribution both form magnetic field and from kinetic motion of particles. These two contribution obey different equations of state, so effectively, we considered relativistic self-similar expansion of a mixture of gasses with different adiabatic indices.
Second, we derived relativistic hodograph and Darboux equations that describe arbitrary one-dimensional motion of magnetized plasma perpendicular to magnetic field. The obtained resulting hodograph and Darboux equations are very powerful: we reduced a system of highly non-linear, relativistic, time dependent equations describing arbitrary (not necessarily self-similar) dynamics of highly magnetized plasma to a single linear differential equation. Using semi-analytical methods we calculated evolution of the flow parameters.
Appendix A Non-Relativistic expansion of hot magnetized plasma into vacuum
In the non-relativistic limit, the equations of one-dimensional transverse MHD read
where is kinetic pressure, is magnetic field divided by , is density is is plasma velocity. The kinetic pressure obeys a polytropic equations of state with index . As the initial condition, we assume that at time plasma occupies region , while at the medium is a vacuum. Initial homogeneous density, magnetic fields and kinetic pressures are , and correspondingly. At time a barrier at is removed and the plasma starts expanding into vacuum while a rarefaction wave propagates into the bulk plasma. In the initial state the Alfvén and sound velocities are
Let us assume that all the quantities depend on self-similar combination . The system (52) then reduces to
where . Eq. (54) can be resolved for velocity ,
and an equation for :
Eq. (56) can be integrated
The constant of integration can be found from the condition that at the front of the rarefaction wave, which propagates with fast velocity in the undisturbed medium and is located at , the plasma density is undisturbed :
In particular, for adiabatic index the previous relations simplify
see Fig. 5. The front of the rarefaction wave propagates with the fast speed and is located at , while the vacuum interface, corresponding to , propagates with the velocity
For unmagnetized plasma Eq. (59) gives